Locally uniform random permutations with large increasing subsequences

TL;DR

This paper explores how the maximum size of increasing subsets in randomly sampled points from certain densities can grow faster than the classical ernik-Vershik result, under specific density conditions, using grid slicing methods.

Contribution

It introduces two sufficient conditions on sampling densities that allow the growth rate of the largest increasing subset to surpass ernik-Vershik's or or N, up to logarithmic factors.

Findings

Growth rate can be any power of N greater than ernik-Vershik's or or N under certain densities.

Methods involve slicing the unit square into grids and analyzing points in each box.

Results extend understanding of increasing subsequences in non-uniform random samples.

Abstract

We investigate the maximal size of an increasing subset among points randomly sampled from certain probability densities. Kerov and Vershik's celebrated result states that the largest increasing subset among $N$ uniformly random points on $[0,1]^2$ has size asymptotically $2\sqrt{N}$. More generally, the order $\Theta(\sqrt{N})$ still holds if the sampling density is continuous. In this paper we exhibit two sufficient conditions on the density to obtain a growth rate equivalent to any given power of $N$ greater than $\sqrt{N}$, up to logarithmic factors. Our proofs use methods of slicing the unit square into appropriate grids, and investigating sampled points appearing in each box.